Structure of extensions of free Araki–Woods factors

Houdayer, Cyril; Trom, Benjamin

Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 473-514.

Nous étudions la structure de produits croisés d’algèbres de von Neumann provenant des actions Bogoljubov de groupes dénombrables sur les facteurs d’Araki–Woods libres. Nous résolvons notamment les questions concernant la factorialité et la classification du type de Connes. Nous donnons également des critères généraux concernant le caractère plein et la solidité forte. Comme application de nos résultats, nous obtenons des exemples de facteurs de type ${\mathrm{III}}_0$ qui sont premiers, sans sous-algèbre de Cartan et qui possèdent une sous-algèbre maximale moyennable abélienne. Nous obtenons aussi une nouvelle classe de facteurs fortement solides de type $\mathrm{III}$ avec des invariants de Connes prescrits et qui ne sont pas isomorphes à des facteurs d’Araki–Woods libres.

We investigate the structure of crossed product von Neumann algebras arising from Bogoljubov actions of countable groups on Shlyakhtenko’s free Araki–Woods factors. Among other results, we settle the questions of factoriality and Connes’ type classification. We moreover provide general criteria regarding fullness and strong solidity. As an application of our main results, we obtain examples of type ${\mathrm{III}}_0$ factors that are prime, have no Cartan subalgebra and possess a maximal amenable abelian subalgebra. We also obtain a new class of strongly solid type $\mathrm{III}$ factors with prescribed Connes’ invariants that are not isomorphic to any free Araki–Woods factors.

Classification : 46L10, 46L40, 46L54, 46L55
Keywords: Free Araki–Woods factors; Fullness; Maximal amenable subalgebras; Popa’s deformation/rigidity theory; Type $\mathrm{III}$ factors; Ultraproduct von Neumann algebras
Mot clés : Facteurs d’Araki–Woods libres ; Facteurs pleins ; Sous-algèbres maximales moyennables ; Théorie déformation/rigidité de Popa ; Facteurs de type $\mathrm{III}$ ; Ultraproduits d’algèbres de von Neumann

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